Networks of geometrically exact beams: Well-posedness and stabilization

Abstract

In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin & Coron in [2] that the zero steady state of this network is exponentially stable for the H 1 and H 2 norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.

Optimisation of Region of Attraction Estimates for the Exponential Stabilisation of the Intrinsic Geometrically Exact Beam Model

“…Theorem 3.1 is proved in [10, Th. 1.5] with more constraints on κ, but the proof is in fact easily adjusted to any positive definite symmetric κ; see also [19,Th. 2.4,Rem.…”

“…One may equivalently study this stability problem for (4) or for its diagonal form (5). Among the methods commonly used to study stability, a so-called H 1 quadratic Lyapunov functional is used in [10,19], namely a functional of the form…”

“…In view of this, our aim in what follows, is to go beyond the specific Lyapunov functional found in [10,19], and look for an optimal functional, in the form of polynomials, such that the bound constraining the size of the initial datum is maximised.…”

Nodal profile control for networks of geometrically exact beams

“…As the form of transmission conditions is an essential aspect in the proof of nodal profile controllability of hyperbolic systems on networks, let us now explain the origin of these conditions for System (11) and especially those of System (15). See also [35] for a more detailed presentation, and for the meaning of the states and coefficients of ( 11) and ( 15).…”

“…One may quickly verify that the matrix A i (x), defined in (8), has only real eigenvalues: six positive ones which are the square roots of the eigenvalues of Θ i (x) (defined in Assumption 1), and six negative ones which are equal to the former but with a minus sign. Furthermore, some computations yield the following lemma whose proof is given in [35,Section 4].…”